L(s) = 1 | − 2-s + 2.70·3-s + 4-s + 3.89·5-s − 2.70·6-s − 2.06·7-s − 8-s + 4.33·9-s − 3.89·10-s + 5.11·11-s + 2.70·12-s + 3.36·13-s + 2.06·14-s + 10.5·15-s + 16-s − 5.20·17-s − 4.33·18-s − 0.951·19-s + 3.89·20-s − 5.58·21-s − 5.11·22-s − 3.39·23-s − 2.70·24-s + 10.1·25-s − 3.36·26-s + 3.60·27-s − 2.06·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.56·3-s + 0.5·4-s + 1.74·5-s − 1.10·6-s − 0.779·7-s − 0.353·8-s + 1.44·9-s − 1.23·10-s + 1.54·11-s + 0.781·12-s + 0.933·13-s + 0.550·14-s + 2.72·15-s + 0.250·16-s − 1.26·17-s − 1.02·18-s − 0.218·19-s + 0.870·20-s − 1.21·21-s − 1.09·22-s − 0.707·23-s − 0.552·24-s + 2.02·25-s − 0.660·26-s + 0.694·27-s − 0.389·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.601817186\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.601817186\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 - 2.70T + 3T^{2} \) |
| 5 | \( 1 - 3.89T + 5T^{2} \) |
| 7 | \( 1 + 2.06T + 7T^{2} \) |
| 11 | \( 1 - 5.11T + 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 + 5.20T + 17T^{2} \) |
| 19 | \( 1 + 0.951T + 19T^{2} \) |
| 23 | \( 1 + 3.39T + 23T^{2} \) |
| 29 | \( 1 - 1.03T + 29T^{2} \) |
| 31 | \( 1 - 5.01T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 - 4.35T + 41T^{2} \) |
| 43 | \( 1 - 0.0596T + 43T^{2} \) |
| 47 | \( 1 + 1.76T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 - 9.42T + 61T^{2} \) |
| 67 | \( 1 + 6.44T + 67T^{2} \) |
| 71 | \( 1 - 1.74T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 4.98T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 5.58T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.872180974964553127730973507747, −8.024382966430377449774243557119, −6.81739146607660737290487784637, −6.47133129896244821914835633540, −5.89958541937215726509111775118, −4.40969497925262168067340953926, −3.55612268363283103103036395638, −2.67968896213900719706598228105, −1.99470987623087225388681073822, −1.25581346138974700983564256287,
1.25581346138974700983564256287, 1.99470987623087225388681073822, 2.67968896213900719706598228105, 3.55612268363283103103036395638, 4.40969497925262168067340953926, 5.89958541937215726509111775118, 6.47133129896244821914835633540, 6.81739146607660737290487784637, 8.024382966430377449774243557119, 8.872180974964553127730973507747